Hierarchy of families of functions

Let $[a,b]$ be a closed, bounded non-degenerate interval.

Let us identify the following family of functions defined on $[a,b]$ .

${\cal F}_{Lip}$ denoting the family of Lipschitz functions.
${\cal F}_{AC}$ denoting the family of absolutely continuous functions.
${\cal F}_{BV}$ denoting the family of functions of bounded variation.
${\cal F}_{Diff}$ denoting the family of functions differentiable almost everywhere.
${\cal F}_{Cont}$ denoting the family of functions continuous almost everywhere.
${\cal F}_{Meas}$ denoting the family of measurable functions.

Then, we have the following set inclusions for the families of functions:

${\cal F}_{Lip} \subseteq {\cal F}_{AC} \subseteq {\cal F}_{BV} \subseteq {\cal F}_{Diff} \subseteq {\cal F}_{Cont} \subseteq {\cal F}_{Meas}$

Furthermore, if we restrict ourselves to bounded functions, we have the following: